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Two Reliability Acceptance Sampling Plans for Items Subject to Wiener Process of Degradation

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Abstract

Traditionally, in reliability acceptance sampling plans, the decision to accept or reject a lot is made by performing the life tests of items. However, when the item’s deterioration is described by a degradation process, it can be made based on the observed deterioration levels of the items obtained from degradation tests. In this paper, two acceptance sampling plans are developed, based on the observation of the deterioration of the items, accumulated on a given period of time. To model the degradation of the items over time, the Wiener process with positive drift is employed. Algorithms to find the parameters of the proposed sampling plans are suggested. Conditionally on the acceptance in the test, the developed sampling plans are shown to improve the reliability performance of the items in the sense that the lifetimes of the items after the reliability sampling test are stochastically larger than those before the test. Also, we compare the two sampling plans both from a technical and economical points of view.

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Abbreviations

Θ:

the frailty r.v. which represents the reliability characteristic of the population

T 𝜃 :

the lifetime of the item with reliability characteristic 𝜃

\(f_{T_{\theta }}(t)\) :

the pdf (probability density function) of T𝜃

{W 𝜃(t), t ≥ 0}:

the conditional degradation process given Θ = 𝜃

{W(t), t ≥ 0}:

the unconditional degradation process

t 0 :

the specified testing duration

{W 1,⋯ .W n}:

the observed degradation levels of the n tested items observed at time t0

κ :

the threshold level of degradation after which the failure of the item occurs

α :

the producer’s risk

β :

the consumer’s risk

(n,c,ξ):

the parameters of Sampling Plan I

\(L_{n,\xi }^{(c)}(\theta )\) :

the lot acceptance probability of Sampling Plan I at 𝜃

(n,ξ):

the parameters of Sampling Plan II

L n,ξ(𝜃):

the lot acceptance probability of Sampling Plan II at 𝜃

References

  • Aslam M, Jun CH (2009) A group acceptance sampling plan for truncated life test having Weibull distribution. J Appl Stat 36:1021–1027

    Article  MathSciNet  MATH  Google Scholar 

  • Aslam M, Jun CH, Ahmad M (2011a) New acceptance sampling plans based on life tests for Birnbaum-Saunders distributions. J Stat Comput Simul 81:461–470

    Article  MathSciNet  MATH  Google Scholar 

  • Aslam M, Jun CH, Ahmad M (2011b) A two-stage group sampling plan based on truncated life tests for a general distribution. J Stat Comput Simul 81:1927–1938

    Article  MathSciNet  MATH  Google Scholar 

  • Aslam M, Balamurali S, Jun CH, Rasool M (2013a) Optimal designing of skip-lot sampling plan of type SkSP-2 with group acceptance sampling plan as reference plan under Burr-type XII distribution. J Stat Comput Simul 83:37–51

    Article  MathSciNet  MATH  Google Scholar 

  • Aslam M, Yen CH, Chang CH, Jun CH (2013b) Multiple states repetitive group sampling plans with process loss consideration. Appl Math Model 7:9063–9075

    Article  MathSciNet  MATH  Google Scholar 

  • Aslam M, Azam M, Jun CH (2013c) A mixed repetitive sampling plan based on process capability index. Appl Math Model 37:10027–10035

    Article  MathSciNet  MATH  Google Scholar 

  • Balakrishnan N, Balasubramanian K, Viveros R (1993) On sampling inspection plans based on the theory of runs. Math Sci 18:113–126

    MathSciNet  MATH  Google Scholar 

  • Balakrishnan N, Leiva V, Lopez J (2007) Acceptance sampling plans from truncated life tests based on the generalized Birnbaum-Saunders distribution. Communications in Statistics - Simulation and Computation 36:643–656

    Article  MathSciNet  MATH  Google Scholar 

  • Blugren W, Hewette J (1973) Double sampling test for hypotheses about the mean of an exponential distribution. Technometrics 22:421–426

    Google Scholar 

  • Cha JH (2016) Analysis of reliability characteristics in the acceptance sampling tests. J Appl Stat 43:1874–1891

    Article  MathSciNet  MATH  Google Scholar 

  • Cha JH (2015) Variables acceptance reliability sampling plan for repairable items. Statistics 49:1141–1156

    Article  MathSciNet  MATH  Google Scholar 

  • Cha JH, Badia FG (2020a) Variables acceptance reliability sampling plan based on degradation test. Statistical Papers https://doi.org/10.1007/s00362-020-01185-1

  • Cha JH, Badia FG (2020b) Variables acceptance reliability sampling plan for items subject to inverse Gaussian degradation process. Journal of Applied Statistics https://doi.org/10.1080/02664763.2020.1723505

  • Chen J, Choy STB, Li KH (2004) Optimal Bayesian sampling acceptance plan with random censoring. Eur J Oper Res 155:683–694

    Article  MathSciNet  MATH  Google Scholar 

  • Doksum KA, Hoyland A (1992) Models for variable-stress accelerated life testing experiments based on Wiener processes and the inverse Gaussian distribution. Technometrics 34:74–82

    Article  MATH  Google Scholar 

  • Epstein B (1954) Truncated life tests in exponential case. Ann Math Stat 25:555–564

    Article  MathSciNet  MATH  Google Scholar 

  • Epstein B, Sobel M (1953) Life testing. Journal of American Statistical Association 48:485–502

    Article  MathSciNet  MATH  Google Scholar 

  • Epstein B, Sobel M (1955) Sequential life test in the exponential case. Ann Math Stat 26:82–93

    Article  MathSciNet  MATH  Google Scholar 

  • Fairbanks K (1988) A two-stage life test for the exponential parameter. Technometrics 30:175–180

    Article  MathSciNet  MATH  Google Scholar 

  • Fernández A J, Pérez-Gonzalez C J, Aslam M, Jun CH (2011) Design of progressively censored group sampling plans for Weibull distributions: An optimization problem. Eur J Oper Res 211:225–532

    Article  MathSciNet  MATH  Google Scholar 

  • Fertig KW, Mann NR (1980) Life test sampling plans for two-parameter Weibull population. Technometrics 22:165–177

    Article  MATH  Google Scholar 

  • Folks JL, Chhikara RS (1978) The inverse Gaussian distribution and its statistical application–A review. Journal of the Royal Statistical Society Series B 40:263–289

    MathSciNet  MATH  Google Scholar 

  • Hu CH, Lee MY, Tang J (2015) Optimum step-stress accelerated degradation test for wiener degradation process under constraints. Eur J Oper Res 241:412–421

    Article  MathSciNet  MATH  Google Scholar 

  • Jin G, Matthews D (2014) Reliability demonstration for long-life products based on degradation testing and a Wiener process model. IEEE Trans Reliab 63:781–797

    Article  Google Scholar 

  • Kahle W, Mercier S, Paroissin C (2016) Degradation processes in reliability. Wiley, Hoboken

    Book  MATH  Google Scholar 

  • Klenke A, Mattner L (2010) Stochastic ordering of classical discrete distributions. Adv Appl Probab 42:392–410

    Article  MathSciNet  MATH  Google Scholar 

  • Lee H, Cha JH (2017) Reliability sampling plan for repairable items following general failure process and its statistical analysis. Statistics 50:1159–1178

    Article  MathSciNet  MATH  Google Scholar 

  • Lu CJ, Meeker MQ (1993) Using degradation measures to estimate a time-to-failure distribution. Technometrics 35:161–171

    Article  MathSciNet  MATH  Google Scholar 

  • Montgomery DC (2012) Introduction to statistical quality control, 7th edn. Wiley, New Jersey

    MATH  Google Scholar 

  • Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, Chichester

    MATH  Google Scholar 

  • Pan D, Liu JB, Cao J (2016) Remaining useful life estimation using an inverse Gaussian degradation model. Neurocomputing 185:64–72

    Article  Google Scholar 

  • Pan D, Liu JB, Huang F, Cao J, Alsaedi A (2017) A Wiener process model with truncated normal distribution for reliability analysis. Applied Mathematical Modelling 50:333–346

    Article  MathSciNet  MATH  Google Scholar 

  • Schneider H (1989) Failure-censored variables sampling plans for lognormal and Weibull distributions. Technometrics 31:199–206

    Article  Google Scholar 

  • Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York

    Book  MATH  Google Scholar 

  • Si XS, Wang W, Hu CH, Chen MY, Zhou DHA (2013) Wiener-process-based degradation model with a recursive filter algorithm for remaining useful life estimation. Mech Syst Signal Process 35:219–237

    Article  Google Scholar 

  • Stephens KS (2001) The handbook of applied acceptance sampling: Plans, procedures and principles. ASQ Quality Press, Milwaukee

    Google Scholar 

  • Tsai TR, Wanbo L (2009) Interval censored sampling plans for the gamma lifetime model. Eur J Oper Res 192:116–124

    Article  MathSciNet  MATH  Google Scholar 

  • Tsai CC, Lin CT, Balakrishnan N (2015) Optimal design for accelerated-stress acceptance test based on Wiener process. IEEE Trans Reliab 64:603–612

    Article  Google Scholar 

  • Van Noortwijk J (2009) A survey of the application of Gamma processes in maintenance. Reliability Engineering & System Safety 94:2–21

    Article  Google Scholar 

  • Wang X (2010) Wiener processes with random effects for degradation data. J Multivar Anal 101:340–351

    Article  MathSciNet  MATH  Google Scholar 

  • Wang X, Xu D (2010) An inverse Gaussian process model for degradation data. Technometrics 52:188–197

    Article  MathSciNet  Google Scholar 

  • Whitmore GA (1995) Lifetime data analysis, pp 1–307

  • Wu CW, Shu MH, Chang YN (2018) Variable-sampling plans based on lifetime-performance index under exponential distribution with censoring and its extensions. Appl Math Model 55:81–93

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Z, Si X, Hu C, Lei Y (2018) Degradation data analysis and remaining useful life estimation: A review on Wiener-process-based methods. Eur J Oper Res 271:775–796

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the reviewer for helpful comments and advices. The work of the first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2019R1A2B5B02069500). The work of the first author was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant Number: 2019R1A6A1A11051177).

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Authors and Affiliations

  1. Department of Statistics, Ewha Womans University, Seoul, 120-750, Republic of Korea

    Ji Hwan Cha

  2. Laboratory of Mathematics and their Applications of Pau (LMAP), University of Pau and Pays of Adour / IPRA / CNRS / E2S UPPA, 64000, Pau, France

    Sophie Mercier

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  1. Ji Hwan Cha
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  2. Sophie Mercier
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Correspondence to Ji Hwan Cha.

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Cha, J.H., Mercier, S. Two Reliability Acceptance Sampling Plans for Items Subject to Wiener Process of Degradation. Methodol Comput Appl Probab 24, 1651–1668 (2022). https://doi.org/10.1007/s11009-021-09879-1

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